Stress Intensity Factor for a Crack Embedded in an Infinite Matrix under the Assumption of Plane Micropolar Elasticity

Abstract

In last few decades, studies on defining mechanical behaviour of materials with significant microstructure have increased drastically. This is due to the lack of compatibility between the experimental data of mechanical behaviour of such materials and obtained results from classical elasticity theory. Moreover, with growing demand and need of using microstructural materials such as polymers and composites in mechanical, physical, and engineering applications, it has become crucial to have a good insight about their behaviour and how to analyse it. As mentioned before, the classical theory cannot provide suitable formulae to model the behaviour of microstructural materials. Thus, a new theory had to be developed in order to describe such materials. The basis of a new elasticity theory, which considers the microstructure of materials, was formed during the 1950s. This theory is known as Cosserat theory of elasticity or micropolar elasticity. An effective way to solve the problems in Micropolar elasticity is to use the boundary integral method. Nevertheless, this method forces some limitations on the properties of boundaries of considerate domains. To be more specific, this method demands more detail to characterize the boundary. By using this method, boundaries can be defined and presented by a twice differentiable curve. As a result, it cannot be applied on domains with reduced boundary smoothness, or the ones containing cracks or cuts. Hence, there is a need of finding methods to define irregular boundaries. There has been some research in this particular area, however this issue has not been completely addressed. In order to overcome this difficulty of defining irregular boundaries, an advanced mathematical approach can be used. This method includes using the distribution setting in Sobolev spaces to formulate the corresponding boundary value problem. The benefit of using this method is finding the appropriate weak solution in terms of integral potentials, which works perfectly for the aforementioned boundaries. In this work, boundary integral equation method has been used to find the integral potentials which are the exact analytical solutions, for the corresponding boundary value problems. Moreover, the boundary element method has been used to approximate these exact solutions numerically. Then these solutions can be applied in many practical engineering problems. As an illustration of the importance of this method, then a crack in a human bone was modeled and solved using these solutions. The bone assumed to follow plane Cosserat elasticity. The stress intensity factor around this crack was calculated and compared to classical analysis. The results approves the high effect of microstructure of the material in stress distribution around the crack.